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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

\(|x+\frac{1}{2}|=|y+\frac{1}{2}|\)

\(=> |x+\frac{1}{2}|^2=|y+\frac{1}{2}|^2\)

\(=> (x+\frac{1}{2})^2=(y+\frac{1}{2})^2\)

\(=> x^2 + x + \frac{1}{4} = y^2 + y + \frac{1}{4}\)

\(=> x^2 - y^2 + x - y = 0\)

\(=> (x-y)(x+y)+(x-y) = 0\)

\(=> (x-y)(x+y+1) = 0\)

\(=> x=y or x+y=-1\)

Condition 1)

Since \(xy < 0\), we have \(x≠y\) and \(x+y = -1\) from the original condition.

Condition 2)

Since \(x > 0\) and \(y < 0\), we have \(x≠y\) and\(x+y = -1\)from the original condition.

Therefore, D is the answer.

By Tip 1), D is most likely to be the answer.

Answer: D

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